Formulation Of Quantum Electrodynamics Research Articles

Absence of spontaneous persistent current for interacting fermions in a one-dimensional mesoscopic ring.

We address the issue of whether a system of interacting electrons confined to a one-dimensional ring can sustain a spontaneous persistent current in the absence of an externally applied flux. The current-current coupling between electrons, describing radiative back-action effects, is exactly treated in the formalism of quantum electrodynamics, where the electrons interact via the exchange of virtual photons. In addition, the instantaneous screened Coulomb potential is taken into account using the Luttinger liquid model including finite-size parity effects. The partition function is calculated exactly, with the result that the system does not possess a spontaneous persistent current. We show that, in the presence of an external flux, the amplitude of the (conventional) persistent current is reduced by quantum fluctuations of the internal transverse electromagnetic field. These corrections can be expressed in terms of the self-induction of the ring and are shown to be of first and higher order in the small dimensionless parameter \ensuremath{\alpha}${\mathit{v}}_{\mathit{F}}^{\mathrm{*}}$/c, where \ensuremath{\alpha} is the fine-structure constant and ${\mathit{v}}_{\mathit{F}}^{\mathrm{*}}$ the Fermi velocity renormalized through Coulomb interactions.

REGULARIZATION OF THE MULTIPOLAR FORM OF QUANTUM ELECTRODYNAMICS

The multipolar form of quantum electrodynamics has been proposed by Power, Zienau et al. It is widely used in nonrelativistic calculations but has the deficiency: its Hamiltonian has a divergent operator term. It is shown that the divergency can be removed by a regularization of the unitary transformation which converts the Coulomb gauge into the multipolar form. The regularized multipolar form is proven to have the same ultraviolet radiative divergencies as the Coulomb gauge electrodynamics. It is also demonstrated that the interaction with soft photons is represented by the usual electric dipole term e qE and interatomic Coulomb interactions persist to be absent.

Photons & Atoms: Introduction to Quantum Electrodynamics

Classical Electrodynamics: The Fundamental Equations and the Dynamical Variables. Lagrangian and Hamiltonian Approach to Electrodynamics. The Standard Lagrangian and the Coulomb Gauge. Quantum Electrodynamics in the Coulomb Gauge. Other Equivalent Formulations of Electrodynamics. Introduction to the Covariant Formulation of Quantum Electrodynamics. References. Index.

QED of resonators in quantum optics

The main features of quantum electrodynamics (QED) of a light-matter system embedded in a dielectric with space-dependent refractive index are developed, and the QED formalism is applied to the description of the quantum-optical action of a resonator-like cavity with in/output coupling and filled with an active medium. In the case of a proper resonator with sufficiently large Q-value the rigorous QED approach naturally leads to the quantum-noise formalisms widely used in a quantum-optical resonator theory. In coarse-graining approximation intracavity-multi-mode photon creation and destruction operators can be introduced, which fulfil quantum Langevin equations with light-matter coupling. Both the time-dependent commutation relations needed to practically handle the Langevin equations and the input-output relations needed to relate the radiation field outside the cavity to the internal field can be established from QED-first principles.

Self-fields in semiclassical radiation theory.

The interaction of an atomic system with an unquantized electromagnetic field is studied by means of the Heisenberg-operator equations of motion. The electromagnetic fields created by the atom are taken into account by assuming that the charge and probability current densities are the actual charge and current source terms in Maxwell's equation. When Ehrenfest's theorem is written to order (1/c${)}^{2}$, the equations of motion are found to have a constant of motion that can be interpreted as stating that the sum of the atomic energy, energy of interaction, and energy stored in the electromagnetic field is constant. This constant of integration can be expressed as an expectation of a new semiclassical Hamiltonian that now includes the effects of the atomic self-fields to order 1/${\mathit{c}}^{2}$. This Hamiltonian is related to the classical Darwin Lagrangian. Since the new approximate Hamiltonian is the sum of an atomic and a field Hamiltonian, it provides a formulation of semiclassical radiation theory that is formally close to the usual formulation of quantum electrodynamics. A Schr\"odinger equation can be derived by applying the variational principle to the expectation of the new Hamiltonian. The result is a nonlinear integro-differential equation, in \ensuremath{\Psi}, which is somewhat similar to Hartree's self-consistent equation for a multielectron atom.When the Heisenberg equations are written to include the next-higher-order terms, 1/${\mathit{c}}^{3}$, it is found that the total energy of the atom and the electromagnetic field in the vicinity of the atom decreases at a rate that is given by the Larmor power formula (2${\mathit{e}}^{2}$/3${\mathit{c}}^{3}$)(d〈v〉/dt${)}^{2}$, where 〈v〉 is the quantum-mechanical expectation of the electron's velocity operator. This approximate formulation of semiclassical radiation theory is applied to semiclassical calculations of atomic radiative shifts. It is then shown that, if some retardation effects are included to all orders of 1/c in the vector potential, the semiclassical analysis provides a formula for radiative energy-level shifts that are time dependent and that contain coefficients similar to the starting point of the Bethe calculation of the Lamb shift. The incorporation of mass renormalization in semiclassical theory is then discussed.

Survey and examination of an electromagnetic vacuum accelerating effect and its astrophysical consequences

A rigorous detailed review of the concepts and calculations behind the acceleration of particles by the electromagnetic zero-point field is presented. The acceleration is enhanced in regions of great size and low particle density. Astrophysical scenarios are suggested and discussed and cosmological possibilities are briefly outlined. It is shown that the acceleration follows both from a semiclassical approach in the manner of Stochastic Electrodynamics and from quantum theory with a zero-point field that most naturally (but not necessarily) may be interpreted as time-symmetric in the manner of the Wheeler-Feynman form of Quantum Electrodynamics. It is shown that both approaches yield essentially identical results.

Relativistic three-fermion wave equation with Coulomb interactions

The variational method, within the Hamiltonian formalism of quantum electrodynamics, is used to derive a relativistic wave equation for a system of three spin- 1 2 fermions, with Coulomb interactions. The equation is shown to reduce to the three-particle Schrödinger equation in the non-relativistic limit, and its perturbative solutions are briefly discussed.

Photons and atoms: introduction to quantum electrodynamics

Classical Electrodynamics: The Fundamental Equations and the Dynamical Variables. Lagrangian and Hamiltonian Approach to Electrodynamics. The Standard Lagrangian and the Coulomb Gauge. Quantum Electrodynamics in the Coulomb Gauge. Other Equivalent Formulations of Electrodynamics. Introduction to the Covariant Formulation of Quantum Electrodynamics. References. Index.

Quantum electrodynamics without the interaction picture

This paper presents a new, axiomatic formulation of quantum electrodynamics which is consistent with Haag's theorem, i.e. it does not require the interaction picture. The method is based on a power series expansion of the interacting fields in the coupling constant, from which the amplitudes for physical processes can be obtained, from first principles, by inspection of the terms and comparison with time-dependent perturbation theory. Up to "tree" level we reproduce the results obtained from Feynman graph analysis.For higher orders, we find divergent integrals, as is the case with other formulations of quantum field theory. Within this method, these infinities can be treated without recourse to the renormalisation procedure, which is an improvement on the traditional methods, but cannot be the final answer.Apart from this, the benefits of this method are (i) the time variable is not eliminated from amplitudes, so it is in principle possible to treat processes other than just scattering and (ii) the notion that the vacuum is invariant and the state of lowest energy in any frame of reference generates no contradiction (there are no vacuum bubbles, for example).

A variational treatment of the relativistic two-fermion bound-state system in quantum electrodynamics

Coupled integral equations for a relativistic two-fermion system of masses m and M are derived variationally within the Hamiltonian formalism of quantum electrodynamics, using an improved ansatz that is sensitive to all terms in the Hamiltonian. The equations are solved approximately to determine the eigenvalues and eigenfunctions, at arbitrary coupling, for various states of the two-particle system.

Relativistic linear combination of Gaussian-type orbitals density functional method based on a two-component formalism with external field projectors

A two-component scalar relativistic method which is based on the no-pair projection operator formalism of quantum electrodynamics has been studied within the framework of local density functional theory. This projection operator formalism, first proposed by Sucher for wave function based methods, provides a stable approach to the relativistic electronic structure problem without the well-known deficiencies of some variational procedures for the Dirac equation. Several approximations of increasing accuracy have been investigated. Besides the method based on free-particle projectors and on external field projectors for the singular nuclear potential, projectors on the full electronic potential are presented here for the first time. Atomic all-electron calculations within this framework give excellent results compared to a fully numerical solution of a scalar relativistic approximation to the corresponding one-particle Dirac problem. Calculations for cerium and for lead are analyzed in detail. About 96% of the relativistic shift of the one-electron energies and about 98% of the relativistic effect on the total energy are covered within this method. The method allows for a rather straightforward generalization to molecular systems with multinuclear potentials.

A model of electron propagation in Zwanziger's formulation of quantum electrodynamics

The author reports a study of the infrared properties of the electron propagator in a framework which uses a gauge-invariant electromagnetic potential. The quantisation of the electromagnetic potential is performed in the Zwanziger formulation (1979) of quantum electrodynamics to avoid the usual infrared divergence. The Kallen-Lehmann representation of the infrared electron propagator in the photon vacuum is shown to be well defined and convergent in this framework. Furthermore, he shows that the probability density of an electron wavepacket propagating in the vacuum decreases like (time)-3, a well known property which is recovered here in quantum electrodynamics.

Quantum electrodynamics based on self-fields, without second quantization: Apparatus dependent contributions to g-2.

Using a formulation of quantum electrodynamics which is not second quantized, but rather based on self-fields, we calculate the energy shifts of an electron bound by a magnetic field in the vicinity of an infinite-plane conductor. We confirm the recent result of Kreuzer that the energy shift arising from the plate-induced change in the magnetic moment, \ensuremath{\Delta}\ensuremath{\mu}/\ensuremath{\mu}=-\ensuremath{\alpha}/4Rm, is exactly canceled by a similar change \ensuremath{\Delta}m/m=-\ensuremath{\alpha}/4Rm in the mass. Thus no change occurs in the spin-precession frequency to order \ensuremath{\alpha}/Rm, in agreement with Brown et al. This cancellation of the two effects resolves an apparent controversy in recent literature over whether such a shift to the spin-precession frequency ${\ensuremath{\omega}}_{s}$ occurs. There is, however, a boundary-induced change in the cyclotron frequency ${\ensuremath{\omega}}_{c}$ which we calculate in the quantum result as \ensuremath{\Delta}${\ensuremath{\omega}}_{c}$/${\ensuremath{\omega}}_{c}$=\ensuremath{\alpha}/8Rm to order \ensuremath{\alpha}. Our method of approach is novel in that it uses only the self-field to compute radiative corrections; there are no vacuum fluctuations.

Quantum electrodynamics based on self-fields, without second quantization: A nonrelativistic calculation of g-2.

Using a formulation of quantum electrodynamics that is not second quantized, but rather based on self-fields, we compute the anomalous magnetic moment of the electron to first order in the fine-structure constant \ensuremath{\alpha}. In the nonrelativistic (NR) case and in the dipole approximation, our result is ${a}_{e}$\ensuremath{\equiv}(g-2)/2=(4\ensuremath{\Lambda}/3m)(\ensuremath{\alpha}/2\ensuremath{\pi}), where \ensuremath{\Lambda} is a positive photon energy cutoff and m the electron mass. A reasonable choice of cutoff, \ensuremath{\Lambda}/m=(3/4, yields the correct sign and magnitude for g-2 namely, ${a}_{e}$=+\ensuremath{\alpha}/2\ensuremath{\pi}. In our formulation the sign of ${a}_{3}$ is correctly positive, independent of cutoff, and the demand that ${a}_{e}$=+\ensuremath{\alpha}/2\ensuremath{\pi} implies a unique value for \ensuremath{\Lambda}. This is in contradistinction to previous NR calculations of ${a}_{e}$ that employ electromagnetic vacuum fluctuations instead of self-fields; in the vacuum fluctuation case the sign of ${a}_{e}$ is cutoff dependent and the equation ${a}_{e}$=\ensuremath{\alpha}/2\ensuremath{\pi} does not have a unique solution in \ensuremath{\Lambda}.

Electric-field-induced two-photon absorption

This paper provides a theoretical treatment of the electrochromic effect of a static electric field on molecular two-photon absorption, based on a full quantum electrodynamical formulation of the interactions. The results include a description of changes in the spectrum resulting both from electro-optical interactions and, in the case of polar fluids, from the anisotropy associated with energetically favored molecular orientations. A detailed consideration of the effect of varying laser polarization and field direction is included in the analysis. The results demonstrate that on application of the field, weak but potentially detectable new lines should appear in the two-photon spectrum. These lines are associated with transitions to vibronic states which would normally require three-photon excitation. The application of the static electric field also results in the selective enhancement of other lines normally present in the two-photon spectrum. The effect may thus offer scope for the accurate determination of the frequency and symmetry character of weak vibronic transitions when they occur under the wing of strongly allowed transitions.

Krein structures for Wightman and Schwinger functions

For quantum field theories that do not satisfy the Wightman positivity condition, a Hilbert space structure condition is proposed, which guarantees a Krein structure for the space of states associated to the Wightman functions. The analogous problem for the Schwinger function is also discussed, as well as the conditions that ensure the analytic continuation of a Krein structure in the Euclidean case and vice versa. The Gupta–Bleuler formulation of free quantum electrodynamics is discussed as an example.

Molecular proximity effects in two-photon absorption

Under laser irradiation, the mutual interaction between molecules is shown to lead to a cooperative nonlinear process which results in entirely new features in the absorption spectrum. In the quantum electrodynamical formalism the coupling between molecular centres, induced by molecular proximity rather than collision, is mediated by means of virtual photon coupling and allows transitions which are forbidden in the absence of such interactions. The manifestation of circular dichroism, potential methods of observation and specific molecular examples are discussed. Attention is drawn to a number of possible resonance enhancement mechanisms and to the dependance of the absorption rate of molecular separation.

Relativistic theory of spontaneous emission.

We derive a formula for the relativistic decay rates in atoms in a formulation of quantum electrodynamics based upon the electron's self-energy. Relativistic Coulomb wave functions are used, the full spin calculation is carried out, and the dipole approximation is not employed. The formula has the correct nonrelativistic limit and is used for calculating the decay rates in hydrogen and muonium for the transitions 2P\ensuremath{\rightarrow}1${S}_{1/2}$ and 2${S}_{1/2}$\ensuremath{\rightarrow}1${S}_{1/2}$. The results for hydrogen are \ensuremath{\Gamma}(2P\ensuremath{\rightarrow}${1S}_{1/2}$)=6.2649\ifmmode\times\else\texttimes\fi{}${10}^{8}$ ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ and \ensuremath{\Gamma}(${2S}_{1/2}$\ensuremath{\rightarrow}${1S}_{1/2}$)=2.4946\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$. Our result for the 2P\ensuremath{\rightarrow}1S transition rate is in perfect agreement with the best nonrelativistic calculations as well as with the results obtained from the best known radiative decay lifetime measurements. As for the hydrogen 2${S}_{1/2}$\ensuremath{\rightarrow}1${S}_{1/2}$ decay rate, the result obtained here is also in good agreement with the best known magnetic dipole calculations. For muonium we get \ensuremath{\Gamma}(2P\ensuremath{\rightarrow}${1S}_{1/2}$)=6.2382\ifmmode\times\else\texttimes\fi{}${10}^{8}$ ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$ and \ensuremath{\Gamma}(${2S}_{1/2}$\ensuremath{\rightarrow}${1S}_{1/2}$)=2.3997\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$.

Quantum-Electrodynamics Based on Self-Energy

We review the basic principles and results of a formulation of quantum-electrodynamics based on the self-energy of the electron, rather than quantized fields. The applications include relativistic spontaneous emission rates, the effect of cavities on Lamb-shift, spontaneous emission and Casimir-Polder forces. We also review the relativistic two-body equations including radiative processes with applications to hydrogen, muonium and positronium spectra.

Fluctuation-dissipation relations. Role of the finiteness of the correlation time. Quantum generalization of Nyquist's formula

The fluctuation-dissipation relations (FDR) in physical systems are studied at all levels of the statistical description. The most general FDR are the relations for the fluctuations of many-body distribution functions. It is pointed out the problem of formulation of FDR is related to the problem of deriving irreversible equations based on the reversible equations of classical and quantum mechanics. The FDR are divided into two classes: 1) FDR for fluctuations with infinite correlation times (collisionless approximation), which correspond to infinitely narrow resonances, and 2) FDR for fluctuations with finite correlation times (collisional approximation). The corresponding spectral densities have finite widths, determined by the collision integrals. The fundamental questions about which different viewpoints have been published in the literature are critically analyzed: 1) the limits of applicability of the Callen-Welton formula and 2) the quantum generalization of Nyquist's formula for the intensity of a Langevin source of oscillatory systems. It is shown that the traditional form of the quantum Nyquist formula, is not well-founded and leads to unphysical consequences. A different expression, used in the literature, for the quantum Nyquist formula is examined. It is not universal, but holds in many important cases. Its region of applicability is determined by the corresponding quantum kinetic equations. The consequences of the two forms of the quantum Nyquist formula, which can be checked experimentally, are studied. The question of the formulation of the quantum Nyquist formula is studied as a part of the general problem of determining the intensity of a Langevin source and the corresponding diffusion coefficient in quantum systems. It arises, in particular, also in quantum electrodynamics in the calculation of the Lamb shift (Sec. 12). In this connection two derivations of Bethe's formula for the Lamb shift are analyzed. It is established that the subtraction formalism of quantum electrodynamics corresponds to the nontraditional form of the quantum Nyquist formula. The exposition is illustrated with many specific examples.